Asymptotic behavior of solutions for the nonlinear Hartree equation involving the fractional Laplacian
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| Format: | Preprint |
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2026
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| _version_ | 1866917525221015552 |
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| author | Borgia, Natalino Cingolani, Silvia Yang, Minbo Zhao, Shunneng |
| author_facet | Borgia, Natalino Cingolani, Silvia Yang, Minbo Zhao, Shunneng |
| contents | In this paper, we investigate the nonlocal problem \begin{equation*}\left\lbrace \begin{aligned} &A_{s} u=(|x|^{-(n-2s)}\ast u^{2_{s}^{\sharp}-1-ε})u^{2_{s}^{\sharp}-2-ε} \quad\quad\hspace{3.5mm} \mbox{in}\hspace{2mm}Ω,\\ &u>0\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{in}\hspace{2mm}Ω,\\ &u=0\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{on}\hspace{2mm}\mathbb{R}^n\setminusΩ, \end{aligned} \right.\end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$, $0<s<1$, $n\in(2s,\min\{6s,n+2s\})$, $ε>0$ small, $2_{s}^{\sharp}-1=(n+2s)/(n-2s)$ and $A_{s}$ stands for the fractional Laplace operator $(-Δ)^{s}$ in $Ω$ with outside zero Dirichlet boundary condition. The above problem is reduced to the subcritical fractional system $$ A_{s}u=u^{2_{s}^{\sharp}-2-ε}v,\hspace{2mm}A_{s}v=u^{2_{s}^{\sharp}-1-ε},\hspace{2mm}u,v>0\hspace{2mm}\mbox{in}\hspace{2mm}Ω\hspace{2mm}\mbox{and}\hspace{2mm}u=(-Δ)^sv=0\hspace{2mm}\mbox{on}\hspace{2mm}\mathbb{R}^n\setminusΩ.$$ For a general domain $Ω$ or domains with convexity, we first prove a uniform $L^1$ bound away from the boundary and a uniform $L^{\infty}$ bound near the boundary for positive solutions to the general fractional Hartree-type PDEs by applying the moving planes method and integral estimates for the convolution term.Among these results, we study the asymptotic behavior of solutions as $ε\rightarrow0$.These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied.Finally,we also establish the corresponding main results for solutions of the fractional Brezis-Nirenberg problem involving critical Hartree-type nonlinearity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23810 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of solutions for the nonlinear Hartree equation involving the fractional Laplacian Borgia, Natalino Cingolani, Silvia Yang, Minbo Zhao, Shunneng Analysis of PDEs In this paper, we investigate the nonlocal problem \begin{equation*}\left\lbrace \begin{aligned} &A_{s} u=(|x|^{-(n-2s)}\ast u^{2_{s}^{\sharp}-1-ε})u^{2_{s}^{\sharp}-2-ε} \quad\quad\hspace{3.5mm} \mbox{in}\hspace{2mm}Ω,\\ &u>0\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{in}\hspace{2mm}Ω,\\ &u=0\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{on}\hspace{2mm}\mathbb{R}^n\setminusΩ, \end{aligned} \right.\end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$, $0<s<1$, $n\in(2s,\min\{6s,n+2s\})$, $ε>0$ small, $2_{s}^{\sharp}-1=(n+2s)/(n-2s)$ and $A_{s}$ stands for the fractional Laplace operator $(-Δ)^{s}$ in $Ω$ with outside zero Dirichlet boundary condition. The above problem is reduced to the subcritical fractional system $$ A_{s}u=u^{2_{s}^{\sharp}-2-ε}v,\hspace{2mm}A_{s}v=u^{2_{s}^{\sharp}-1-ε},\hspace{2mm}u,v>0\hspace{2mm}\mbox{in}\hspace{2mm}Ω\hspace{2mm}\mbox{and}\hspace{2mm}u=(-Δ)^sv=0\hspace{2mm}\mbox{on}\hspace{2mm}\mathbb{R}^n\setminusΩ.$$ For a general domain $Ω$ or domains with convexity, we first prove a uniform $L^1$ bound away from the boundary and a uniform $L^{\infty}$ bound near the boundary for positive solutions to the general fractional Hartree-type PDEs by applying the moving planes method and integral estimates for the convolution term.Among these results, we study the asymptotic behavior of solutions as $ε\rightarrow0$.These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied.Finally,we also establish the corresponding main results for solutions of the fractional Brezis-Nirenberg problem involving critical Hartree-type nonlinearity. |
| title | Asymptotic behavior of solutions for the nonlinear Hartree equation involving the fractional Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.23810 |